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Time-Correlation Functions

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Presentation Transcript

Organization

- Time Correlation Function: Definitions and Properties
- Linear Response Theory :
- Fluctuation-Dissipation Theorem
- Onsager’s Regression Hypothesis
- Response Functions

- Chemical Kinetics
- Transport Properties
- Self-diffusivity
- Ionic Conductivity
- Viscosity

- Absorption of Electromagnetic Radiation
- Space-time Correlation Functions

Time Correlation Functions

- Time-dependent trajectory of a classical system:
- Since the classical system is deterministic, a time-dependent
quantity can be written as :

- Correlation function as atime average over a trajectory:

- Time-correlation functions can be written as ensemble averages by summing over all possible initial conditions:
Probability of observing a microstate

- Limiting behaviour
- Alternative definition of time-correlation function in terms of deviations of time-dependent properties from mean values.

- Stationarity for systems with continuous interparticle forces, TCFs must be even functions of the time delay:
- Time-derivative with respect to time origins must be zero
- Short-time expansion of autocorrelation functions

Typical velocity autocorrelation function forces,

Zero slope at origin

Rebound from

Solvent cage

D. Chandler

Small Deviations from Equilibrium: forces, Classical Linear Response Theory

- Apply a weak perturbing field f to the system that couples to some physical property of the system
- Electric field/ionic motion
- Electromagnetic radiation/charges or molecular dipoles

System allowed to relax freely

Equilibrium

established

System prepared

in non-equilibrium

state by applying

perturbing field f

B(t)

D. Chandler

time=0

Linear Response Theory (contd.) forces,

Let the time-dependent perturbation be such that

At t=0, the probability of observing a configuration:

How will the observed value of a quantity B(t) change with time when the perturbation is turned off at time t=0?

Time-dependent value of

B for a given set of initial

conditions

Integrate over initial

conditions of perturbed

system at t=0

Linear Response Theory (contd.) forces,

Consider the effect of perturbations only upto first-order:

D. Chandler, Introduction to Modern Statistical Mechanics

For t>0 forces, , the observed value of B will be given by

Multiply the numerator and denominator by (1/Q) where Q is the partition function of the unperturbed system

Denominator

Numerator: forces,

Time-dependent behaviour of B:

Onsager’s regression hypothesis forces,

The relaxation of macroscopic non-equilibrium disturbances is governed by the same dynamics as the regression of spontaneous microscopic fluctuations in the equilibrium system

Macroscopic

relaxation

Equilibrium

Time-correlation

function

Response Functions forces,

For a weak perturbation , we can define a response function:

The response of the system to an impulsive perturbation:

To correspond to the linear response situation studied earlier:

- Provides an alternative route to evaluate the time-dependent response as an integral over a time-correlation function

Transport Properties response as an integral over a time-correlation function

- Flux=-transport coefficient X gradient
- Non-equilibrium MD: create a perturbation and watch the time-dependent response
- Equilibrium MD: measure the time-correlation function

Self-diffusivity response as an integral over a time-correlation function

Consider an external field that couples to the position of a tagged particle such that

The steady state velocity of the tagged particle will then be:

Can one identify the mobility, as defined below, with the macroscopic velocity:

Fick’s Law:

Flux of diffusing species= Diffusivity X Concentration gradient

Combining with the equation of continuity derived by imposing conservation of mass of tagged particles, gives :

Diffusion Equation

If the original concentration profile is a delta-function, then the concentration profile at a later time (t) will be a d-dimensional Gaussian:

Consider the second-moment of one-dimensional distribution:

Einstein relation

By writing the displacement as

one can show that

This definition of self-diffusivity will be the same as that of the mobility derived from linear reponse theory

Ionic Conductivity then the concentration profile at a later time (t) will be a

Consider an external electric field Ex applied to an ionic melt. Under steady state conditions, the system will develop a net current:

The ionic conductivity per unit volume, s, will be defined by:

Effect of the external field on the Hamiltonian:

When the field is switched off at t=0, the current will decay towards the zero value characteristic of the unperturbed system.

Rate of change of net

dipole moment

Charge on

particle i

velocity

particle i

To apply the relation: then the concentration profile at a later time (t) will be a

to compute the time-dependent decay of the current, we set:

to obtain:

Conductivity per unit volume will then be given by:

Collective Transport Properties then the concentration profile at a later time (t) will be a Silica (6000K, 3.0 g/cc)

Ionic Conductivity

Viscosity

Linear Response Theory and Spectroscopy then the concentration profile at a later time (t) will be a

- Let f(t) be a periodic, monochromatic disturbance:
- Time-dependent energy:
- Rate of absorption of energy:

Linear Response Theory and Spectroscopy (contd.) then the concentration profile at a later time (t) will be a

- Time-dependent value of A reflects response to applied field:
- The average rate of absorption or energy dissipation is given by:
- For a periodic field:

Linear Response Theory and Spectroscopy (contd.) then the concentration profile at a later time (t) will be a

- Fourier transform of response function is defined as:
- Compute average rate of absorption of energy over one time period T=2p/w

Linear Response Theory and Spectroscopy (contd.) then the concentration profile at a later time (t) will be a

- Using linear response theory
- Absorption spectrum

Simple Harmonic Oscillator then the concentration profile at a later time (t) will be a

- Let the quantity A coupled to the periodic perturbation obey SHO dynamics:
- Time-dependence of A:
- Absorption spectrum

Infrared Absorption by a Dilute Gas of Polar Molecules then the concentration profile at a later time (t) will be a

- X-component of total dipole moment will couple to oscillating electric field .
- Independent dipole approximation:
- Perturbed Hamiltonian:
- Change in dipole moment with time:
- Thermal distribution of angular velocities, P(w), will be reflected in absorption profile

Spectroscopic Techniques then the concentration profile at a later time (t) will be a

Space-Time Correlation Functions: then the concentration profile at a later time (t) will be a Neutron Scattering Experiments

Number density at a point r at a time t:

Conservation of particle number:

Van Hove Correlation Function for a homogeneous fluid:

Space-Time Correlation Functions (contd) then the concentration profile at a later time (t) will be a

Can divide the double summation into two parts:

Self contribution

Distinct contribution

Fourier transform of the number density

Space-Time Correlation Functions (contd.) then the concentration profile at a later time (t) will be a

Intermediate scattering function

Static structure factor

Dynamic structure factor

Sum rule

References then the concentration profile at a later time (t) will be a

- D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications
- D. C. Rapaport, The Art of Molecular Dynamics Simulation (Details of how to implement algorithms for molecular systems)
- M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (SHAKE, RATTLE, Ewald subroutines)
- Haile, Molecular Dynamics Simulation: Elementary Methods
- D. Chandler, Introduction to Modern Statistical Mechanics (Linear Response Theory)
- D. A. McQuarrie, Statistical Mechanics (Spectroscopic Properties)
- J.-P. Hansen and I. R. McDonald, The Theory of Simple Liquids (Almost everything)

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